Seasonality in a Two-strain
Competition Model for Trypanosoma
Cruzi Transmission
Catherine Rogers
Christopher M. Kribs-Zaleta
Technical Report 2013-02
http://www.uta.edu/math/preprint/
Seasonality in a two-strain competition model for
Trypanosoma cruzi transmission
Catherine Rogers and Christopher M. Kribs-Zaleta
May 6, 2013
Abstract
Previous studies have observed both strains of T. cruzi coexisting in some sylvatic
host populations (namely woodrats) despite cross-immunity precluding coinfections
in hosts. In this study we explore the possible role of seasonality in demographic
parameters explaining this coexistence, using a system of ordinary differential equations
to model demographic and epidemiological processes.
We analyze seasonal T. cruzi transmission dynamics in three sylvatic cycles—
raccoons with T. sanguisuga, woodrats with T. sanguisuga, and woodrats with T.
gerstaeckeri, calculating the invasion reproduction and replacement numbers (IRNs)
for each strain at each point in time over a one-year period to see which one ”wins”.
Results of numerical analysis indicate there is a clear winner for each cycle, but seasonality alone does not support coexistence.
1
Introduction
Seasonality plays an important role in the dynamics of vector-borne diseases. Altizer
et al. (2006) [1] review examples from human and wildlife disease systems to show the
challenges inherent in understanding the mechanisms and impacts of seasonal drivers
in the environment. Upon an extensive review, they cover many different ways seasonality changes the dynamics in hosts and vectors. The effects include periodic changes in
the biology of hosts and pathogens or vectors. This means seasonal changes in growth,
death, and feeding rates can alter the outcome of whether a disease can spread successfully or not [1]. Chavez and Pascual (2006) looked into how climate cycles forecast
Leishmaniasis cases. They used monthly data spanning ten years of cases of Leishmaniasis in Costa Rica to come up with a model. Their results showed that there is
in fact a peak time of outbreaks of Leishmaniasis cases [6]. These studies and many
more show that seasonality is important in effecting changes in vector-borne disease
outbreaks.
Trypanosoma cruzi is a parasite responsible for infecting 10-12 million persons in
Latin America with Chagas disease, and has a large impact on heart disease cases
in this area of the world since it mimics heart disease. T. cruzi is transmitted by a
vector genus called Triatoma. There have been cases reported in the United States
with humans, but the sylvatic transmission cycles are the ones which keep the parasite
going, with vectors moving towards human populated areas in search of new food
sources [22]. In our study, we investigate the sylvatic transmission of the disease
between the vectors and sylvatic hosts such as raccoons and woodrats. Many studies
show that the two vector species found with these specific hosts in the southern U.S.
are T. sanguisuga and T. gerstaeckeri, and T. gerstaeckeri is only found with woodrats
while T. sanguisuga is found with both hosts [22].
1
Roellig et al. did a study about the two different strains of T. cruzi found in the
United States. Their results showed strain I was found with opossums, triatomine
vectors, humans, and rhesus macaques; they found strain IV (formerly IIa) with one
primate, a few raccoons, and placental mammalian isolates [37]. Charles et al. (2012)
found that woodrats tested positive for both strains (I and IV) of T. cruzi [5]. One
important question is whether one strain outcompetes the other. Hall et al. (2010)
did a study that compares vertical transmission of the two different strains of T. cruzi
in mice. Their results showed greater virulence of strain I and breeding experiments
demonstrated a reciprocal relationship between virulence and the frequency of vertical
transmission, with pups born to type IV infected female testing positive at twice the
frequency as those infected with type I. This would imply that strain IV wins in vertical
transmission competition [10].
Competition between the two strains is the main focus of this paper. Many studies have been done on strain competition using math modeling. Feng et al. (2002)
did a study on the competition of the two strains of Tuberculosis (drug-sensitive and
drug-resistant). They wanted to analyze the effects of variable periods of latency on
the transmission dynamics of TB at the population level. Their results showed that
with limited access to medical care, the drug-resistant TB will be a major problem
[8]. McLean and Nowak (1992) study strain competition, with a math model, of drug
resistant and drug sensitive strains of HIV. In their model, they state the behavior is
best understood by separately looking at the two issues of competition between the
strains of virus and host-parasite interactions between infected and uninfected cells.
With considering the two different cases, it coincides with separate consideration of
long-term equilibrium behavior of the model and its shorter-term, dynamic behavior.
With these two cases considered separately competition between virus strains is best
considered in terms of reproductive fitness, which is characterized as the basic reproductive rate. They show that competition between strains of virus is the important
factor that determines which type of the virus will eventually start to grow during the
course of the drug treatment, but host-parasite interactions determine which type will
dominate when viral resurgence occurs [29].
Kribs-Zaleta and Mubayi (2012) study strain competition of T. cruzi extensively. In
their paper they review literature to come up with average parameter values for each
host (raccoons and woodrats) and each vector (T. sanguisuga and T. gerstaeckeri.
Then they use a system of ODEs to analyze the results. In their analysis they look at
the basic reproductive numbers (BRNs), and invasion reproductive numbers (IRNs).
Their results conclude that it is possible for either strain (strain I or IV) to win in a
population. The way this is possible is based on each strain’s degree of adaptation.
So if we have very little vertical and oral transmission, strain IV must adapt to oral
and vertical transmission in order to win the competition in raccoons. Under the same
assumptions strain I wins the competition in woodrats, but greater vertical or oral
transmission rates allow strain IV to win [22].
One of the possible explanations for two different strains winning the competition is
seasonality. For instance, woodrats’ and raccoons’ breeding seasons are different times
of the year. So one strain adapted to vertical transmission might have an advantage
during peak host breeding season, while another strain adapted to stercorarian transmission might have an advantage during the peak vector feeding season. There are a
few ways to model seasonality; the way we will model seasonality is seasonal forcing.
This basically applies a periodic function to the infection rate to allow for periods of
high and low with the average in the middle. We will use the model described in
Kribs-Zaleta and Mubayi (2012) [22] to apply seasonality and analyze the results. We
will evaluate the role seasonality plays in strain competition. Since the principle of
competitive exclusion says species competing for the same resources cannot coexist if
other ecological factors are constant, there must be an explanation such as seasonality
2
to explain why we have coexistence. Meaning, they never win at the same time, but
if you break the year up into smaller time frames, one wins over the other for small
periods of time and vice versa.
First shown is an extensive literature review on seasonality, math models, and how
the two are used together. Then we describe the model from [22] and explain how
we modified it to incorporate seasonality. Once this is complete, all of the seasonal
parameters are defined with the dates of the maxima, minima, and ranges for each,
and where we found this information. After a parameter synthesis, we explain how we
analyzed the model, and broke it up into square wave analysis and continuous wave
analysis. From there we suggest some conclusions and how this study can be taken
further.
2
Literature Review
Seasonality plays a prominent role in birth rate, death rate, and feeding rate for the
hosts and the vectors. Based on this, we hypothesize that it will play a role in one strain
of T. cruzi winning over the other. The review covers three topics: (1) How seasonality
may affect ecological models, (2) How seasonality affects epidemic models, and (3) How
mathematical models incorporate seasonality. The first section gives an overview of
the importance of seasonality on basic ecology. The second section ties seasonality to
epidemic models since we are working with that type of model. The final section goes
into detail on how seasonality is incorporated into these models mathematically. Upon
reviewing these papers, we use the most applicable processes to incorporate seasonality
into our model.
One field study [27] on two different flocks of birds showed seasonality and forest fragmentation play a role in species richness, size, and stability. They studied a
heterogeneous flock and understory flock in nine Atlantic forest fragments in Minas
Gerais, southeastern Brazil. The two different seasons they used were the rainy and
dry seasons. In the heterogeneous flocks, size, richness, and stability were significantly
affected, but only size was affected by the season change for the understory flocks. They
hypothesize that these changes may be due to resident species joining flocks during the
dry season only. Also, richness in the heterogeneous flocks may be due to seasonal
changes in flood availability, or reproductive activities, which influence the motivation
for species to join flocks [27].
In their data analyses they started the first wet season on the 30th of March and
kept it going for six months, then started the dry season. They used multiple regression
models to examine the effects of fragment area, fragment isolation, and the number of
relief types in fragments on these flock characteristics for each season separately. The
six regression models showed a strong multi-collinearity between fragment area and
number of relief types, not allowing the use of both variables in the regression models.
In the end, they found forest fragment area influenced the richness, size, total number
of species, and stability of heterogeneous flocks during both dry and rainy seasons [27].
We will look at a research paper[14] that examined the effects of light and prey
availability on nocturnal, lunar and seasonal activity on tropical nightjars. We want
to see if/how seasonality plays a role in the dynamics of tropical living creatures.
They studied nightjars in the West African bush savannah. They examined how light
regime and prey abundance affect activity across different temporal scales. They used
a statistical model that generalized linear models employing maximum quasi-likelihood
estimation of the response mean-variance relationship [14].
They started the wet season on March 30th and applied it for six months. They
found potential prey availability was highest at dusk, and lower at dawn and during
the night. Intuitively, the most intensive foraging for the nightjars was at dusk, then
3
slightly less at dawn, and least during the night. They also found that nocturnal
foraging was positively correlated with lunar light levels and completely ceased once
the light got below a certain level. They also increased twilight foraging activity
during new moon periods, compensating for a shorter nocturnal foraging window at
night. Seasonality affected prey availability, which peaked shortly after onset of the
wet season and slowly decreased over the following four months [14].
Since our study uses mammals for the hosts, we will look at a paper [23] displaying how seasonality affects homeotherms population. The paper chosen examines how
seasonality affects the body size of mammals. Basically they gathered data from many
resources and analyzed the significance of large body sized mammals for enhancing fasting endurance. They used a dynamical nonlinear system of ODEs and incorporated
seasonality by expanding Wunder’s 1985 model to form one single equation that predicts an animal’s total rate of energy use as a function of body mass, body temperature,
and ambient temperature. The data they collected showed many correlations between
body size and seasonality within species of mammals. Due to high mortality when
resources are scarce, surviving individuals benefit from low competition and abundant
resources during the growth season. This will favor rapid growth and larger individuals
due to enhanced survivorship and in some species increased fecundity. This implies
seasonality may produce two outcomes of body-size evolution: (1) reduced densitydependent competition because of seasonal high mortality and (2) increased fasting
endurance for individuals of larger size. Their results showed the scaling of fasting
endurance favors large body size [23].
Another study [17] researches the effect seasonality has on ecological models. They
used a model on a bivoltine population to test the effects of seasonality. They used a
dynamical nonlinear system of ODEs that incorporates seasonality by using population
and capacity parameters to describe sinusoidal functions. They used, r: potential
rate for the population increasing at density zero, and K: carrying capacity to vary
seasonally. Normally those are fixed in a differential equation model. Fixing those
values would imply a constant population, which is not true in reality. The periods
used were two six-month periods starting in fall and spring. Their results showed large
seasonality is inevitably destabilizing, but mild seasonality may have a pronounced
stabilizing effect. Also, seasonality allows for coexistence of alternate stable states
[17].
There are many studies on how seasonality affects all different types of populations.
These were just a few examples. The point they make is seasonality plays a large role
in changing important factors in many species of living organisms, and we apply them
to the vectors and hosts in our model. This is the reason we felt it is very important to
incorporate seasonality into our model. The previous examples look at how seasonality
affects things like growth and food rates. Since our model looks at two different strains
of a parasite, we will now look at studies that examine how seasonality changes the
spread of a disease, looking at epidemic models incorporating seasonality.
This first study [1] is a large review of examples from human and wildlife disease systems to show the challenges in understanding the mechanics and impacts of seasonal
environment drivers. It summarizes the importance of seasonality in mathematical
models. One way they suggest analyzing seasonality in an epidemic model is to pinpoint the relative importance of seasonal drivers that increase or decrease R0, which is
an important first step in understanding their roles in the dynamics of infectious diseases. There are many reasons why seasonality is important: (1) Temperature changes
affect vector disease transmission by possibly killing off vectors if temperatures get too
low, (2) Seasonal changes affect larvae development, (3) Seasons predict breeding seasons which affect reproduction, which increases susceptible populations, (4) Seasonal
changes affect levels of immunity, e.g. rodents’, birds’, and humans’ immune systems
are weakened in the winter [1].
4
One method of incorporating seasonality is seasonal forcing in host social behavior
and aggregation. Hosts’ birth, death, feeding, and other rates vary seasonally. The
way Altizer et al. (2006) incorporates seasonal forcing is to integrate cosine into the
function
β(t) = β0 (1 + β1 cos(2πt)).
A lot of models integrate seasonality in phenomenological ways, either by dividing time
into discrete intervals or adding time delays into continuous-time models [1].
Another study [15] looks at incorporating seasonality into a generic model using seasonal forcing. This study uses three key features common to many biological systems:
temporal forcing, stochasticity, and nonlinearity, these features are used to analyze the
seasonal forcing method. They used a simple SIR model compared to real data, and
examine how these three factors potentially change to produce a range of complicated
dynamics. Models of childhood diseases incorporate seasonal variation in contact rate
due to higher levels of mixing while school is in session [15].
The periods they chose were based on holidays for school kids. Their periodic
function, which is +1 during school terms and -1 during holidays was based on holiday periods of days 356-6, 100-115, 200-251, 300-307.The diseases they focus on are
measles, whooping cough, and rubella. They combine seasonal forcing approximated
by a sinusoidal function and a square wave to come up with a function
β(t) = β0 (1 + β1 )T erm(t)
where the term is when kids are in school or when kids are on holiday, β0 is the contact
rate, and β1 is seasonality. They conclude SIR models are incredibly accurate at
predicting possible outbreaks if done properly. The dynamics of a seasonally forced SIR
model is best considered with two attractors in this case, one for term-time infection
rate, and one for holiday infection rate. Very small differences in infection rates can
lead to annual dynamics [15].
The next study we look at [32] models the effect of weather and climate change on
malaria transmission. Studies in the past have shown that malaria transmission is sensitive to environmental conditions, which is why they chose this disease to model with
seasonality. The reason for this study is to show how dynamic process-based mathematical models can give important insight into the effects of climate change on malaria
transmission. They analyzed a simple dynamical systems model that provided information on the effects of varying temperature and rainfall simultaneously on mosquito
population dynamics, malaria invasion, persistence and local seasonal extinction, and
the impact of seasonality on transmission [32].
Their objective was to adopt a coarse-grained dynamic model to explore the general
dynamical impact of climatically driven systems on malaria transmission. They used
the spatial nature of their model output to get the result of parameterization by local
values of temperature and rainfall. They use monthly periods with levels of rainfall
and temperature both varying. Their results showed extinction was more strongly
dependent on rainfall than temperature. They also identified a window of temperature
where endemic transmission and the rate of spread in a totally susceptible region is
optimized [32].
Another study [39] on the transmission of malaria had similar results. The interesting factor in this paper is they collected detailed data to get the correct climate
estimates, using The Hadley Centre global climate model (HAD CM3). The main
climate components they incorporated were temperature and rainfall. They collected
data from 15 sites for which published malaria seasonal profiles existed. They used
a dynamical system of nonlinear ODEs with two monthly variables (moving average
temperature and rainfall) and three yearly variables (minimum temperature, standard
deviation of average monthly temperature, and existence of a catalyst month). They
5
used monthly periods varying the temperature and rainfall. Then they validated the
model by analyzing 6284 laboratory-confirmed parasite-ratio surveys and compared
the results to their model. Their results showed the effect of possible climate change
suggesting that a prolonged transmission season is as important as geographical expansion in correct assessment of the effects of changes in transmission patterns. Their
model provides a valid baseline against which climate scenarios can be assessed and
interventions planned [39].
Another study [7] used seasonal forcing to investigate the general conditions that
promote the sub-harmonic resonance behavior that may lead to multi-annual cycles
in a general malaria dynamical model. They used two complementary approaches to
bifurcation analyses to show that resonance is promoted by processes decreasing the
time of the infectious period and that sub-harmonic cycles are favored in situations
with strong seasonality in transmission. They introduce seasonality into their model
by allowing the vector-host ratio to vary sinusoidally over the year. The analyses given
demonstrated that the likelihood of eliciting sub-harmonic resonance in malaria under
periodic annual forcing is increased by: (1) lengthening the period individuals remain
immune from the scale of months to years; (2) shortening the length of the infectious
period, possibly by medical intervention; and (3) reducing the transmission rate in
high transmission areas, for example, by decreasing the vectorial capacity. These were
modeled using average delays, meaning ODEs [7].
These are a few of the examples out there that incorporate seasonality into an
epidemic model to analyze the results. Since the main idea is to model what could
possibly happen in the future, to be able to efficiently prepare for a possible outbreak
of a disease, we want the model to be as realistic as possible. In doing so, we want to
add seasonality, since it plays a role in the environment anywhere in the world. Lastly,
we will look at the details in how some of the mathematical modeling papers actually
incorporate the seasonality into their models. This last section will summarize two
papers on the process of incorporating seasonality.
Schwartz (1985) describes mathematical models in a general way by incorporating
seasonality with seasonal forcing. The model shows bistable behavior for a fixed set
of parameters. They use the basic SEIR model and incorporate seasonality into the
contact rate by making it a periodic function with cosine. Then they look at seasonality
and the multiple recurrent epidemics that occur. Once this is completed, they have
periodic orbits, and then they look into the basins of attraction for these orbits. Once
they get this information, they can determine the predictability of an epidemic model.
This shows that numerically the geometry of basins of attraction for small and large
amplitude recurrent epidemics for a seasonally driven SEIR epidemic is important in
understanding epidemic outbreaks [38].
The final study [11] builds an age-structured model of the dynamics of tick populations to examine how changes in average temperature and temperature variability
affect seasonal patterns of tick activity and transmission of tick-borne diseases. Development rate of ticks is temperature dependent, from eggs into larvae, engorged larvae
developing into nymphs, engorged nymphs developing into adults, and engorged adults
producing eggs. This means the growth of the ticks is temperature dependent and
therefore seasonally affected. They used temperature data from a location that is
known for frequent outbreaks of a tick-borne disease and the growth stages as their
periods for their model. They also used a dynamical nonlinear stochastic differential
equation system with probability [11].
In order to predict the number of ticks of a given life stage present each week of
the year they created an age-structured matrix model that tracks the survival and
growth of each weekly cohort of ticks through the lifecycle stages. They also included
an effect of temperature on the weekly host finding probability for the ticks, so if
the temperature got below a certain level, the ticks had zero probability that week of
6
finding a host. Diapause is common for ticks; they incorporated this into the model by
specifying that a certain fraction of questing larvae, nymphs, and adults that obtain a
blood meal in the months August-December delay the onset of interstadial development
until the first of the following year. The model records the development time lags over
a series of years to create seasonal patterns in tick abundance that reflect inter-annual
temperature change [11].
In conclusion, seasonality should definitely be considered when trying to model any
biological population. The ecology studies prove that seasonality does play a large
role in changing factors as important as R0. Mathematical models of epidemics are
using seasonality and temperature variation more and more. From this review, we see
different ways to mathematically incorporate seasonality, which parameters can change
results based on season, which periods they chose to use and why, how seasonality was
modeled, and their results based on seasonality. One method used to model seasonality
is adding multiple classes to the basic SIR model; another, more common method, is
to use a periodic function for the infection rate. In our model, we will first describe
seasonally varying parameters using square waves and analyze the results through the
lens of the invasion reproductive numbers, which are defined in the analysis section.
Then we will form continuous wave functions to match our periods and compare those
results.
3
3.1
Models
Baseline model for infection dynamics
The model being used for this study to describe infection dynamics was created by
Kribs-Zaleta and Mubayi (2012) [22]. This is a model to describe competition between
T. cruzi I and IV (denoted strains 1 and 2 in the model). The first step will be to
summarize the model from Kribs-Zaleta and Mubayi (2012) [22], and then we will
explain how we will adapt the model to incorporate seasonality.
One important aspect to notice is how infection rate is calculated. The three modes
of infection to hosts are vertical transmission, stercorarian, and predation. All these
modes of transmission are possibly affected by seasonality. For vertical transmission,
we only consider female hosts, and assume a proportion pj (j=1,2) of hosts infected
with strain j give birth to infected young. When vectors obtain a blood meal from
an infected host, the rate at which that vector gets infected is cvj , while the rate an
infected vector infects a host is chj for strain type j (j = 1,2). Finally, a proportion
ρj of hosts that consume an infected vector with strain j become infected with that
strain.
Normally models use standard incidence or mass action incidence to calculate infection rates. In this model there are multiple infection rates and the three contact-based
rates use piecewise linear (Holling type I) saturation [18, 19, 20]. Kribs-Zaleta and
Mubayi [22] go into detail about how the infection rates are found. In the model Q
is defined to be the vector-host population density ratio Q = Nv /Nh and is used to
calculate infectious contact rates. The contact rates are defined as follows:
Q
, 1),
Qv
1/Q
Qv
cvj (Q) = βvj min(
, 1) = βvj min( , 1),
1/Qv
Q
Q
Eh (Q) = H min(
, 1),
Qh
chj (Q) = βhj min(
with maximum values βhj , βvj (j = 1, 2) and H respectively [22].
7
(3.1)
(3.2)
(3.3)
Table 1: Variables and notation for sylvatic T. cruzi transmission model.
Variable
Sh (t)
Ih1 (t)
Ih2 (t)
Sv (t)
Iv1 (t)
Iv2 (t)
Q
chj (Q, Qv )
cvj (Q, Qv )
Eh (Q, Qh )
Meaning
Density of uninfected hosts
Density of hosts infected with T. cruzi I
Density of hosts infected with T. cruzi IV
Density of uninfected vectors
Density of vectors infected with T. cruzi I
Density of vectors infected with T. cruzi IV
Vector-host population density ratio (Nv /Nh )
Strain j stercorarian infection rate
Strain j vector infection rate
Per-host predation rate
Units
hosts/area
hosts/area
hosts/area
vectors/area
vectors/area
vectors/area
vectors/hosts
1/time
1/time
vectors/host/time
Table 2: Parameter definitions for sylvatic T. cruzi cycles, with annual average values taken
from [22]. The few parameters that stay constant will not have a ”t”.
Parm.
rh (t)
rv (t)
µh (t)
µv (t)
N*h
Kh
N*v
Kv
Qh (t)
Qv (t)
βhmax (t)
βv (t)
pmax
H(t)
ρmax
ρ
Definition
Growth rate for hosts
Growth rate for vectors
Mortality rate for hosts
Mortality rate for vectors
(Equilibrium) host population density
Carrying capacity for hosts
(Equilibrium) vector population density
Carrying capacity for vectors
Threshold vector-host density ratio for predation
Threshold vector-host density ratio for bloodmeals
Maximum stercorarian infection rate
Vector infection rate
Maximum vertical transmission proportion
(Maximum) per-host predation rate
Maximum proportion of hosts infected after
consuming an infected vector
Estimated proportion of hosts infected after
consuming an infected vector
Units
per year
per year
per year
per year
hosts/acre
hosts/acre
vectors/acre
vectors/acre
vectors/host
vectors/host
per year
per year
dimensionless
vectors/host/yr
R/S
0.90
33
0.40
0.271
0.080
0.14
128
129
10
100
0.525
9.67
0.15
1
W/G
1.8
100
1
0.562
9.3
21
128
129
10
100
13.5
1.59
0.375
1
host per vector
1
1
host per vector
0.177
0.177
All of the infection rates could possibly be affected when seasonality is incorporated
into the model. The stercorarian infection rate could be affected due to vectors going
into a state similar to diapause. Upon going into this state, they feed less, and in
turn, transmit the disease less. For vertical transmission, births of hosts and vectors
vary seasonally. This means that hosts giving birth to infected young could be more
prominent during the birth season for that host. There is no vertical transmission for
the vectors because the parasite only resides in the gut of the infected vectors [22].
Finally, for predation, the hosts feed on insects more during certain times of the year;
this would decrease predation rates when hosts are feeding on other types of food
sources.
The total vector density is Nv = Sv + Iv1 + Iv2 , same applies to total host density
Nh . The total vector birth rate bv is
bv (N ) = rv N (1 − N/Kv ),
(3.4)
and same for host birth bh , where h is representative of either woodrats (W) or raccoons
(R) and v is T. sanguisuga (S) or T. gerstaeckeri (G). The resulting system of ordinary
equations is as follows:
8
Sh0 (t)
=
p1 Ih1 (t) + p2 Ih2 (t)
1−
Nh (t)
bh (Nh (t)) − µh (t)Sh (t)
Iv1 (t)
Nv (t)
Iv2 (t)
− [ch2 (Q(t)) + ρ2 (t)Eh (Q(t))]Sh (t)
,
Nv (t)
Ih1 (t)
Iv1 (t)
0
Ih1
(t) = p1
bh (Nh (t)) + [ch1 (Q(t)) + ρ1 (t)Eh (Q(t))]Sh (t)
− µh (t)Ih1 (t),
Nh (t)
Nv (t)
Iv2 (t)
Ih2 (t)
0
bh (Nh (t)) + [ch2 (Q(t)) + ρ2 (t)Eh (Q(t))]Sh (t)
− µh (t)Ih2 (t),
Ih2
(t) = p2
Nh (t)
Nv (t)
Sv (t)
Sv0 (t) = bv (Nv (t)) − µv (t)Sv (t) − Eh (Q(t))Nh (t)
Nv (t)
Iv2 (t)
Ih1 (t)
− cv2 (Q(t))Sv (t)
,
− cv1 (Q(t))Sv (t)
Nh (t)
Nh (t)
Ih1 (t)
Iv1 (t)
0
Iv1
(t) = cv1 (Q(t))Sv (t)
− µv (t)Iv1 (t) − Eh (Q(t))Nh (t)
,
Nh (t)
Nv (t)
Iv2 (t)
Ih2 (t)
0
− µv (t)Iv2 (t) − Eh (Q(t))Nh (t)
.
(3.5)
Iv2
(t) = cv2 (Q(t))Sv (t)
Nh (t)
Nv (t)
− [ch1 (Q(t)) + ρ1 (t)Eh (Q(t))] Sh (t)
Tables 1 and 2 display state variables, parameters, and notations. The parameters given were taken from [22]. Kribs-Zaleta and Mubayi [22] introduce
adaptive trade-off. Trade-off is described as how each strain adapts in order to
improve or decrease the chance of infection for each strain. These adaptations
change the parasite’s capacity for the different modes of infection transmission
such as stercorarian (βhj ), vertical (pj ), and oral (ρj ). We want to see how tradeoff will affect the different transmission modes, so we change the parameters in
order to see the effects clearly. The trade-off variables x, y, and z measure the different infectivities applied to stercorarian, vertical, and oral transmission. If any
of the variables has a value of 1, this means that the transmission is fully adapted
towards that given mode. If any of the variables have a value of 0, this means
that the transmission has no adaptation to transmit the infection in that mode.
What was just written is described here in math terms: for a given strain j, we
can write βhj = βhmax xj , pj = pmax yj , and ρj = ρmax zj , with xj , yj , zj ∈ [0, 1]
[22].
The model from [22] assumes that trade-off only appears with stercorarian
and vertical transmission. They define a relationship between the variables as
follows, if x equals 1, then y must equal 0, and vice versa. There has not been
data published to confirm whether oral infectivity also adapts so they consider
both cases. One case is where oral transmissibility does not depend on the
stercorarian-vertical adaptation, meaning zj is fixed, and the other case is oral
transmissibility does depend on vertical adaptation, meaning zj = yj (j=1,2).
The final assumption made is that when there is just one mode of vector infection,
and the parasites are in a different life stage in the vectors than in the hosts, then
vector infection rates are not changed by the adaptation trade-off, meaning βv1
= βv2 = βv . For our model we use the x1 = 0.755 and x2 = 0.431 [22].
Many of the parameters described above in Table 2 vary seasonally. The
growth rates for raccoons and woodrats depend on what time of year they breed.
9
This also applies to the vectors. The mortality rate, and predation/stercorarian
rates also vary seasonally due to temperatures making different food sources
more or less abundant. All of these parameters are defined and explained in
detail in the parameter synthesis. The seasonally varying parameters will first
be described in terms of square waves (piecewise constant) for a period of one
year to analyze the results. Once we complete the square wave analysis, we will
apply to sine waves the same varying parameters to get more realistic results
since they vary continuously with time.
3.2
Square-wave model and parameter synthesis
The first model we look at is a square-wave model. Here we break the year up
into eleven periods and apply a square-wave to one period (one year) and analyze
the results. In order to do so, we need parameters for each host and vector to
apply to each period. In the parameter synthesis below we describe in detail
where the parameters came from, how we calculated them, the maximum and
minimum times of year, and the range for each value.
There are not many studies done on how seasonal changes may affect transmission of the two different strains of T. cruzi. The processes with the greatest
seasonal fluctuations in our model are birth, death, stercorarian contact rates
for each vector species, and predation rates for the host species. Using previous
biological studies done on each host and vector species, we took the maximum
and minimum rates for the summer and winter months and applied these values
to our model. In the following paragraphs we explain in detail where these values
came from.
Some of the information found is not absolute seasonal rates. For rates for
which only relative seasonal rates are given, we can derive the absolute seasonal
rates from annual averages by constructing a function of period one year with
average value of one and then multiply by the annual average rate to get the
maximum and minimum seasonal rates; we will call this the seasonal averaging
method. For simplicity we will take each period to last 6 months and begin/end
halfway between the dates of the two extrema when the values are symmetric
about the average.
Otherwise, when they are not symmetric about the average we adjust in the
following way: we use the average, maximum, and minimum to get the durations
to be correct. In general, average = max ∗ x + min(1 − x) and solve for x to
determine the periods. For example, if .30 was a max, .15 was a minimum and,
.225 was the average:
(0.30x) + (0.15(1 − x)) = 0.225.
Once you solve for x, you multiply that by 365 to get the duration for the
maximum period, and 1 − x gives the duration for the minimum period.
In order to determine where to start these durations, we use literature cited
below to get the maximum and minimum values and times of year the demographic characteristics occur, such as birth, mortality, and feeding rates for the
hosts and vectors. Once we get the maximum and minimum with the times of
year, you place half of the duration before the maximum and half after. Also, in
some cases further minor adjustments were made to starting dates in order that
the model can have periods no shorter than two weeks. Once you put all the
10
host and vector periods together, some of the overlap was less than two weeks so
the days were adjusted minimally to fit our model. In some cases, dates within
one or two days of each other are aligned for simplicity.
Raccoons give birth once a year with the breeding season peaking early March
as the temperatures begin to rise. Once the breeding takes place, an average of
3.4 young per pregnant female are born during April or May with a gestation
period of 63 days. 50% of female raccoons are giving birth in early May, and
none in November [3, 25, 30, 35]. We multiply:
g = 3.4
1 litter
1.5s.m.yr. 1fem.racc.
raccoons
∗
∗
∗
= 0.9/yr.
litter
fem.racc.*s.m.yr.
2.5yr.
racc.
where s.m.yr stands for sexually mature year(s), the third component is the
proportion of a female raccoon’s life of which she can reproduce, and the fourth
component eliminates the males since only females reproduce, to get the average
growth rate per year [21].
The average annual growth rate is 0.9/year, from Kribs-Zaletas (2010) review.
Using the above formula, seasonal averaging method, and raccoon peak breeding
to be early May, we calculate female raccoons to have a maximum growth rate
of 1.8/year peaking on May 7th with a range from February 5th to August 6th.
The minimum growth rate is 0 on November 6th with a range from August 7th
to February 4th [21].
Raccoons have varying mortality rates due to hunting and climate. We average the seasonal maximum and minimum mortality rates from Prange et al.
(2003) and Zeveloff (2002) to get 32.7% for the maximum on April 1st and 13% for
the minimum on September 30th. The annual average mortality rate is 0.4/year
[21]. Taking these averages and using the seasonal averaging method we find
raccoons have a maximum mortality rate of 0.57/year on April 2nd with a range
from January to June, and minimum mortality rate of 0.23/year on October 1st
with a range from July to December [41, 34, 21].
One of the ways we can measure contact rates between raccoons and vectors
is with the predation rate for raccoons. A study in a research refuge in Maryland
showed that raccoons do eat insects at different rates based on time of year, and
the reason is due to availability of diet needed, which is the same reasoning as
the study done in Texas. Since Maryland has a different climate from Texas
and therefore different vector actions, we will not use their rates [24]. The study
done on the food habits of raccoons in eastern Texas states that insects are an
important part of a raccoon’s diet, and gives feeding rates for raccoons. Raccoons
eat the most insects in early February, 15%, and the least in early November, 4%
[2].
The annual average max predation rate for raccoons is 1/year [21]. Using
the seasonal averaging method, this average from Kribs-Zaleta (2010), and the
seasonal information from Baker et al. (1945), we estimate that the maximum
predation rate for raccoons occurs on March 15th with a range from December
16th to June 15th and a scaling factor of 1.58/year. The minimum predation rate
for raccoons occurs on September 15th with a range from June 16th to December
15th and a scaling factor of 0.42/year. We apply this scaling factor to both the
maximum per host predation rate (H) and threshold vector-host density ratio
for predation (Qh ), both default values are found in Table 2 [2, 24, 21].
Based on a study done in western Texas, 48% of woodrats are breeding at the
end of July and they found no woodrats breeding mid December. Also, Kribs11
Zaleta (2010) cites information on the age of sexual maturity for woodrats and
calculates growth rates. Using this information with the formula from KribsZaleta (2010) we multiply:
g = 2.73 ∗ 2 ∗ 0.6 ∗ 0.5 = 1.8/yr.
where 2.73 is the average litter size, two is the average number of litters per
year, and 60% is the amount of time a female woodrat is sexually mature. This
gives an average growth rate of 1.8/year [21]. Using the formula and information
above we use the seasonal averaging method to calculate the maximum growth
rate to be 3.6/year on July 5th with a range from April 1st to October 4th, the
minimum growth rate is 0 on January 4th with a range from October 5th to
April 4th [26, 40, 21].
For the woodrat mortality rate we will use the rate of disappearance from the
study by Raun in 1966. Unfortunately mortality is extremely difficult to measure
in a wild population. Raun (1966) estimates that woodrats have a 60% mortality
rate in the winter peaking in early January. Kribs-Zaleta’s (2010) information
gives the average mortality rate for woodrats as 1/year. Using Raun’s (1966)
estimate, and the seasonal averaging method, we found the maximum mortality
rate to be 1.60/year on January 4th with a range from October to March 30th,
and the minimum mortality to be 0.40/year on July 5th with a range from March
31st to September [36, 21].
Woodrats also eat insects; again we will use this as a measure of predation
between woodrats and vectors. Woodrats’ first choice of food is green vegetation. When that is unavailable, they eat what they can find, like insects [36, 4].
A study of Neotoma magister in the central Appalachians showed that in early
February 0.4% of the woodrats’ total diet was insects, and in early November it
was 3.0%. The annual average maximum predation rate is estimated to approximately be 1/year [21]. Using this information and the seasonal averaging method,
we calculate that the maximum scaling factor to be 1.76/year on September 15th
with a range from June 15th to December 15th. The minimum scaling factor is
0.24/year on March 15th with a range from December 16th to June 14th, again
these scaling factors affect H and Qh from Table 2 [36, 4, 21].
This following will cover growth rate for the vectors. There are very few
articles written about the number of eggs laid by the two species we focus on
T. sanguisuga and T. gerstaeckeri in field conditions. When studying other
species, and in laboratory conditions, fecundity estimates from infestans Klug of
the genus Triatoma average 5.4 eggs per female per week with a maximum of 17
[?]. Since this is a different species from the two we are studying, and in different
conditions, we will use the data given by Pippin (1970).
Pippin’s (1970) study in southern Texas of T. sanguisuga, p.41 Table 11,
found that they lay the most eggs in the middle of June with 131 eggs
131eggs 12months
∗
= 1572eggs/yr.
1month
1year
which gives you a maximum of 1572 eggs per year. The corresponding hatch rate
(from the same table) is 70.9%. The least eggs are laid in early December with 0
(Pippin 1970). Using the above information, the data from Kribs-Zaleta (2010),
and his formula we multiply:
g=
1572eggs
1.44s.m.yrs.
hatch
adult 1female
∗
∗ 0.71
∗ 0.60
∗
= 131/yr.
s.m.yr.*female
3.69yr.
egg
hatch 2adults
12
where 1.44 years is the expected adult lifetime, 3.69 years is the average total
lifetime, and 60% being the rate of survivorship to adulthood for both species.
The average growth rate is 33/year [21]. In this case our maxima and minima are
not symmetric about the average, so our ranges will not be mirror symmetric.
We calculated the maximum growth rate to be 131/year, but in order to have
the peak value be in the middle of the range, we need to adjust it to 124/year on
June 20th with a range from May 2nd to August 7th, minimum growth rate to
be 0 on December 19th with a range from August 8th to May 1st [31, 12, 33, 21].
The same study also examined T. gerstaeckeri, in the field, and found the
most eggs were produced in June with 188 eggs, which gives a maximum of 2256
eggs per year with a hatch rate of 68%. The maximum number of eggs that
hatched was in May with 138 eggs, but since June fits the 6-month range in our
table the best, we chose June since it is only a difference of 10 with 128 hatching.
The least number of eggs laid were in December with 6 eggs and hatch rate of
0%. We can use the previous data, along with the formula from Kribs-Zaleta
(2010) and multiply:
g=
0.82s.m.yrs.
hatch
adult 1female
2256eggs
∗
∗ 0.68
∗ 0.60
∗
= 259/yr.
s.m.yr.*female
1.78yr.
egg
hatch 2adults
The average growth rate was found from Kribs-Zaleta’s (2010) paper to be
100/year, since the values are not symmetric about the average the date ranges
are not mirror symmetric. We calculate the maximum growth rate to be 259/year
on May 28th with a range from March 19th to August 6th and a minimum
growth rate of 0 on November 27th with a range from August 7th to March 18th
[28, 21, 33].
Relative mortality rates for these vectors are not possible to find thus far.
Pippin (1970) states that vectors act in a similar manner as insects in diapause
in the wintertime; we will use this to help estimate mortality rate. A study done
by Kiyomitsu and Tadafumi (1998) found mortality rates of two species of insects,
Orius sauteri and O. minutus, which go through diapause in order to survive, to
be 50% less in the winter than the annual average. Using Kribs-Zaleta’s (2010)
mortality average, we take 50% less in the winter, and 50% more in the summer
to get the maximum and minimum rates [21, 16].
The average mortality rate for T. sanguisuga is 0.271/year [21]. We calculated
the maximum mortality rate to be 0.41/year on August 1st with a range from
May to October and minimum mortality to be 0.14/year on February 3rd with a
range from November to April [21, 33]. For T. gerstaeckeri, the average mortality
rate is 0.562/year [21]. We calculated the maximum mortality to be 0.84/year,
and the minimum mortality to be 0.28/year. The dates and ranges are the same
as for T. sanguisuga [16, 21].
For T. sanguisuga and T. gerstaeckeri, using a chart from Grundemann (1947)
that displays the specific dates collected, number of vectors captured, and the
percent of vectors engorged, we can model the stercorarian contact rate based on
time of year. The average rate for T. sanguisuga is 0.53/year, and the average
for T. gerstaeckeri is 13.5/year [21]. We use the seasonal averaging method, and
calculate the scaling rate using the average to find the scaling factor. Using Grundemann’s (1947) Table 1, we calculate the maximum and minimum dates, Grundemann’s (1947) results showed a maximum of 44 captured and 47.7% engorged
in the summer and minimum of 3 captured and 0 engorged for T. sanguisuga in
the winter. Since, to date, there are no published papers on the seasonal feeding
13
Table 3: Seasonally varying parameters. In the mathematica file we actually used half days
where necessary to make each period exactly correct.
Species
Raccoon
Max.
Min.
Avg.
Woodrat
T. sang.
Max.
Min.
Avg.
Max.
Min.
Avg.
T. gerst.
Max.
Min.
Avg.
Growth
Mortality
Scaling factor
for feeding
Value Peak Range
(1/yr) (days) (days)
Value Peak Range
(1/yr) (days) (days)
Value
Peak Range
(days) (days)
1.8
0
0.9
3.6
0
1.8
124
0
33
259
0
100
0.57
0.23
0.4
1.60
0.40
0.4
0.41
0.14
0.4
0.84
0.28
0.4
1.58
0.42
1
1.76
0.24
1
1.92
0.08
1
1.96
0.04
1
74
258
350-166
167-349
258
74
167-349
350-166
186
4
95-277
278-94
186
4
95-277
278-94
127
310
36-218
219-35
186
4
95-277
278-94
171
353
122-218
219-121
148
331
78-218
219-77
92
274
1-182
183-365
4
186
278-94
95-277
213
34
122-304
305-121
213
34
122-304
305-121
rate of T. gerstaeckeri, we will use the results found by Grundemann (1947) and
the annual averages from Kribs-Zaleta’s 2010 paper.
For T. sanguisuga, the maximum scaling factor is 1.92/year on July 5th with a
range from April to September, and a minimum rate of 0.08/year on January 4th
with a range from October to March. For T. gerstaeckeri we find the maximum
rate to be 1.96/year and a minimum of 0.04/year; the dates are the same as for
T. sanguisuga. These scaling factors are applied to all the β’s and Qv ’s from
Table 2 [9, 21]. Table 3 shows all the seasonally varying parameter maximums
and minimums with their ranges, and table 4 defines all the 11 different phases
for the square-wave piecewise functions.
14
Table 4: Period lengths and period values for square-wave piecewise functions. The equilibrium
population density ratio Q∗ is given for each period and each transmission cycle (v/h); values of
0 or ∞ reflect periods where one or both species’ growth rates are 0.
Period
Days
rr (t)
µr (t)
ψr (t)
rw (t)
µw (t)
ψw (t)
rs (t)
µs (t)
φs (t)
rg (t)
µg (t)
φg (t)
Q∗s/r
Q∗s/w
Q∗g/w
1
1-35
0
0.57
1.58
0
1.6
0.24
0
0.14
0.04
0
0.28
0.04
∞
∞
∞
2
36-77
1.8
0.57
1.58
0
1.6
0.24
0
0.14
0.04
0
0.28
0.04
0
∞
∞
3
78-94
1.8
0.57
1.58
0
1.6
0.24
0
0.14
0.04
259
0.28
0.04
0
∞
∞
4
95-121
1.8
0.57
1.58
3.6
0.4
0.24
131
0.14
1.06
259
0.28
1.96
0
0
14.02
5
122-166
1.8
0.57
1.58
3.6
0.4
0.24
131
0.41
1.06
259
0.84
1.96
1585
15.92
12.09
6
167-182
1.8
0.57
0.42
3.6
0.4
1.76
131
0.41
1.06
259
0.84
1.96
1565
13.79
10.84
7
183-218
1.8
0.23
0.42
3.6
0.4
1.76
0
0.41
1.06
259
0.84
1.96
1522
12.4
10.04
8
219-277
0
0.23
0.42
3.6
0.4
1.76
0
0.41
1.06
0
0.84
1.96
0
0
0
9
278-304
0
0.23
0.42
0
1.6
1.76
0
0.41
0.04
0
0.84
0.04
0
∞
∞
10
305-349
0
0.23
1.58
0
1.6
1.76
0
0.14
0.04
0
0.28
0.04
∞
∞
∞
11
350-365
0
0.23
1.58
0
1.6
0.24
0
0.14
0.04
0
0.28
0.04
∞
∞
∞
The equilibrium values for Q based on constant annual-average parameters
are as follows:
Q∗s/r = 1600, Q∗s/w = 13.76, Q∗g/w = 13.76.
You find these by dividing Nv∗ /Nh∗ . The moment-to-moment values calculated
for Q(t) for each period are not symmetric about the equilibrium value. The
reason for this is periods with no growth for vectors really make the population
density drop significantly even if growth is doubled during other periods and
the asymmetric graph in Figure 1 really demonstrates that with T. sanguisuga
and raccoons. In Mathematica, when you use 0 for the minimum growth rate of
vectors and 66/yr for the maximum value each for 6 months of the year, instead
of a constant 33/yr (the annual average), we observed the average value for Nv (t)
drops from 127.93 to 125.7 vectors/acre. So in our model, the drop is exaggerated
because our growth rate is an asymmetric function.
15
1650
1600
1550
1500
1450
1400
2
4
6
8
10
Figure 1: Comparison of Q(t) for square-wave model (asymptotically periodic curve) with
Q(t) for constant-parameter model (asymptotically constant curve).
3.3
Sine-wave model
The second model we use is a continuous sine-wave model. We use the parameters
found from the parameter synthesis and apply them to continuous functions.
Since we have the maximum, minimum, average, and phase length, we can use
these to come up with the functions. For all the parameter values with high
and low phases of six months, we can apply a sine-wave function to them by
adjusting the amplitude and shifting it up in order to match our values. For the
two parameters that do not have six month ranges, we modified a trig function
to match where the phases change from low to high as close as possible and
most importantly have the average values match the baseline model [22]. For the
asymmetric functions, we use a trig function composed with:
f (t) = t + sin(2πt),
(3.6)
where = 0.0844 for T. sanguisuga and = 0.02469 for T. gerstaeckeri. The
growth for T. sanguisuga was composed with the f function twice, while the
growth for T. gerstaeckeri was composed with f three times. The reason for
this is because we first made the average work out exactly, then we found the
value for epsilon that matched the transition from low to high periods as close as
possible. We use these functions below to solve the system of ODEs and analyze
the results. Table 5 shows all the functions used to define seasonally varying
parameters for the sine-wave model.
The continuous growth functions for the vectors do not transition from high
to low phases at the exact time as the square-wave model, meaning the ranges
are not exactly where we have them in the square-wave model. Even though this
changes the phase lengths from the square-wave model, we make sure to preserve
the annual average rate. In both cases, it makes the ”high” phase is longer by a
few days, which is supported by Pippin in table 11 [33].
16
Table 5: Continuous functions for sine-wave model, using f (t) as defined in the text.
Raccoon
Growth
Mortality
Feeding
Woodrat
Growth
Mortality
Feeding
Sanguisuga
Growth
Mortality
Feeding
Gerstaeckeri
Growth
Mortality
Feeding
4
4.1
0.9 + 0.9 cos(2π(t − 127.25/365))
0.4 + 0.17 cos(2π(t − 92.25/365))
1 + 0.58 cos(2π(t − 75.25/365))
1.8 + 1.8 cos(2π(t − 186.25/365))
1 + 0.6 cos(2π(t − 5.25/365))
1 + 0.76 cos(2π(t − 258.25/365))
62 + 62 cos(2πf (f (t − 171.5/365)))
0.27 + 0.135 cos(2π(t − 213.25/365))
1 + 0.92 cos(2π(t − 186.25/365))
129.5 + 129.5 cos(2πf (f (f (t − 148.5/365))))
0.56 + 0.28 cos(2π(t − 213.25/365))
1 + 0.96 cos(2π(t − 186.25/365))
Analysis/Results
Baseline model for infection dynamics
Kribs-Zaleta and Mubayi (2012) conclude that either strain of T. cruzi can win
the competition. Based on their estimates of infection-related rates and how the
different strains adapt, they conclude with their estimates of infection-related
rates and each strain’s degree of adaptation, with very moderate amounts of vertical and oral transmission rates, strain IV must adapt to oral transmission along
with vertical transmission in order for strain IV to be able to win with raccoons.
If you make the same assumptions, strain I wins in woodrats, but with greater
vertical or oral transmission rate, then strain IV will win. Based on observations
in the field, both strains are found in woodrat populations, which would imply
that slightly greater vertical or oral transmission in woodrats puts the competing
strain on close to even ground. This takes place where local random effects and
small amounts of interaction between other woodrat populations close by allow
different parasite strains to invade a population [22].
Charles et al. (2012) [5] found both strains of the parasite in woodrat populations; this would imply that slightly greater oral or vertical transmission rates
in woodrats would put both strains on an almost even playing ground. In their
analysis, they concluded that oral transmission being a second contact process
saturating in the vector-host ratio will operate as a considerable mediator for
the competition. This can be concluded even when both of the strains are orally
transmitted at the same level. They imply it would be a good idea to extend
the study to explore ways both strains could coexist since T. cruzi I is found in
very small amounts in raccoons and larger amounts in woodrats in transmission
cycles where strain IV is also found [22].
In the following analysis we will see if/when seasonality allows this to happen.
In order to analyze which strain wins for each period we compare the basic
reproductive numbers (BRNs) R1 and R2 for strains 1 and 2 respectively and the
e1 and R
e2 for the respective strains. BRNs
invasion reproductive numbers (IRNs) R
are important in math models that incorporate diseases. The reason for this is
because they give the mean number of secondary infections that come about
17
from a single infected individual in a population that is completely susceptible.
You calculate the BRNs once the system has reached a stable state, meaning
equilibrium. The IRNs give the mean number of secondary infections brought on
by the given strain from a single infected individual that is put into a population
where the other strain is already native [42]. In our model, we will be calculating
the replacement numbers instead of the basic reproductive number. Replacement
numbers are ”defined as the average number of secondary infections produced by
a typical infective during the entire period of infectiousness” [13]. So when we
write IRN, we imply invasion replacement number. The following formulas are
given for constant parameters from [22], as follows:

R1 =
1
p1 +
2
s



s
e
e
e
e
βh1 βv1 
βh2 βv2 
1
, R2 = p2 + p22 + 4
, (4.1)
p21 + 4
µh µ
ev
2
µh µ
ev
s



s
e
e
e
e
βh1 βv1  e
1
βh2 βv2 
p21 + 4(1 − p2 )
, R2 = p2 + p22 + 4(1 − p1 )
.
e
e
2
βh2 βv2
βeh1 βev1
and

e 1 = 1 p1 +
R
2
(4.2)
where
βehj = chj (Q∗ ) + ρj Eh (Q∗ ) and βevj = cvj (Q∗ )
(4.3)
and chj (Q), cvj (Q), and Eh (Q), are as given in (3.1), (3.2), (3.3). In our model,
we will apply the same formulas, except our parameters are changing with time.
So when we calculate the BRNs and IRNs, they will be changing depending
on which period of time we are in for the square-wave model and continuously
changing for the sine-wave model. This will allow us to determine if certain
strains win at specific times of the year.
4.2
4.2.1
Square-wave model
Equilibrium vector-host ratios
We had to break this up into a few different cases since hosts and vectors have
a growth rate of zero during some periods. Case 1 is when both growth rates
are greater than zero, then we apply the formulas from above (4.1 and 4.2), and
e1 , and R
e2 . Case
we get values for all different reproductive numbers, R1 , R2 , R
2 is when the growth rate for vectors is greater than zero, but the growth rate
for hosts is zero, so Q(t) → ∞ meaning Q(t) > Qh and Q(t) > Qv . We can only
e1 and R
e2 . We accept this because BRNs are calculated at equilibrium
calculate R
normally. In our model, since each period IRN or BRN is calculated for only a
short amount of time, it is not long enough to reach equilibrium. So instead we
focus on the IRNs using equations (4.2) in the limit with Q∗ → ∞. Case 3 is
similar to case 2, except growth rate for vectors is zero, and growth rate for hosts
is greater than zero, so Q(t) → 0 meaning Q(t) < Qh and Q(t) < Qv . Again, we
e1 and R
e2 using equations (4.2) in the limit with Q∗ → 0.
only get values for R
18
When both growth rates are zero, we have to take a few different steps. We
use the equations from Kribs-Zaleta and Mubayi [22] :
Nh0 (t) = bh (Nh (t)) − µh Nh (t),
Nv0 (t) = bv (Nv (t)) − µv Nv (t) − Eh (Q(t))Nh (t).
(4.4)
(4.5)
with bh = bv = 0 to calculate Nv (t) and Nh (t). Upon solving Nh0 (t) we get
Nh (t) = Nh (0)e−µh t . In order to solve for Nv (t), we need to break it up into two
main cases. The reason for this is because Eh is defined as H ∗ min( NNh Qv h , 1).
Case 4 is when Q > Qh ⇒ Nv > Qh Nh and Case 5 is when Q < Qh ⇒
Nv < Qh Nh . Case 4 gives
Nv0 = −µv Nv (t) − HNh (t) whence
−µv t
e
− e−µh t
−µv t
.
Nv (t) = Nv (0)e
+ HNh (0)
µv − µh
(4.6)
(4.7)
Once you get these calculations we need to check a few things. For Case 4 we
check to see at what point in time, if ever, does it go to Case 5 by having Nv ≤
Qh Nh . Once you do some algebra you see the parameter ranges or conditions
that change the results are:
4a. µh < µv ,
4b. µv < µh < µv +
4c. µv +
H
Q0
H
,
Q0
< µh < µv +
4d. µh > µv +
H
,
Qh
H
.
Qh
Since each case starts out with an assumption on whether Q(t) < Qh or
Q(t) > Qh , we will apply the same assumption to Q0 . Meaning if Q(t) < Qh ,
then Q0 < Qh . We do this because we want to start in the case we are working
with. We will apply the four sub cases to Case 4.
Case 4a is when µh < µv (⇒ µh < µv + QH0 < µv + QHh ); we rewrite Nv (t) ≤
Qh Nh (t) as
−(µv −µh )t
e
−1
−(µv −µh )t
Q0 e
+H
≤ Qh
(4.8)
µv − µh
⇒ (µv − µh )Q0 e−(µv −µh )t + He−(µv −µh )t − H ≤ Qh (µv − µh )
Qh (µv − µh ) + H
⇒ −(µv − µh )t ≤ ln
Q (µ − µh ) + H
0 v
1
Qh (µv − µh ) + H
⇒t≥
ln
−(µv − µh )
Q0 (µv − µh ) + H
(4.9)
(4.10)
(4.11)
At this point in time, Q(t) will cross below Qh when the above conditions are
satisfied.
For Case 4b, when µv < µh < µv + QH0 < µv + QHh , we rewrite Nv (t) ≤ Qh Nh (t)
as
−(µv −µh )t
e
−1
−(µv −µh )t
≤ Qh
(4.12)
Q0 e
+H
µv − µh
⇒ (µv − µh )Q0 e−(µv −µh )t + He−(µv −µh )t − H ≥ Qh (µv − µh )
1
Qh (µv − µh ) + H
⇒t≥
ln
−(µv − µh )
Q0 (µv − µh ) + H
19
(4.13)
(4.14)
At this point in time, Q(t) will cross below Qh when the above conditions are
satisfied.
For Case 4c, when µv + QH0 < µh < µv + QHh , we rewrite Nv (t) ≤ Qh Nh (t) as
−(µv −µh )t
e−(µv −µh )t − 1
µv − µh
≤ Qh
(4.15)
⇒ (µv − µh )Q0 e−(µv −µh )t + He−(µv −µh )t − H ≥ Qh (µv − µh )
(4.16)
⇒ e−(µv −µh )t (Q0 (µv − µh ) + H) ≥ Qh (µv − µh ) + H
Qh (µv − µh ) + H
−(µv −µh )t
⇒e
≤
Q0 (µv − µh ) + H
(4.17)
Q0 e
+H
(4.18)
Because of the initial conditions, this implies a power of e is less than or equal
to a negative number, so with these conditions Q(t) stays above Qh .
For Case 4d, when µh > µv + QHh > µv + QH0 , we rewrite Nv (t) ≤ Qh Nh (t) as
−(µv −µh )t
e−(µv −µh )t − 1
µv − µh
≤ Qh
(4.19)
⇒ (µv − µh )Q0 e−(µv −µh )t + He−(µv −µh )t − H ≥ Qh (µv − µh )
(4.20)
Q0 e
+H
−(µv −µh )t
⇒e
(Q0 (µv − µh ) + H) ≥ Qh (µv − µh ) + H
1
Qh (µv − µh ) + H
ln
⇒t≤
−(µv − µh )
Q0 (µv − µh ) + H
(4.21)
(4.22)
This case makes t ≤ (negative value) which is not possible in our model, so
again, with these conditions Q(t) stays above Qh . To summarize, in Case 4,
Q(t) ≥ Qh ⇒ Q0 ≥ Qh , when µh < µv + QH0 then Q(t) will cross below Qh . When
µh > µv + QHh then Q(t) stays above Qh .
Case 5 gives
Nv
0
Nv = −µv Nv − H
whence
(4.23)
Qh
Nv = Nv (0)e
−(µv +
H
Qh
)t
(4.24)
For Case 5 we only need to consider two sub cases:
5a. µh < µv +
H
,
Qh
5b. µh > µv +
H
.
Qh
Case 5a is when µh < µv +
H
;
Qh
we rewrite Nv (t) ≥ Qh Nh (t) as
−(µv + QH )t
≥ Qh e−µh t
H
Qh
⇒ (µh − µv −
)t ≥ ln
Qh
Q
0
1
Qh
⇒t≤
ln
H
Q0
(µh − µv − Qh )
Q0 e
(4.25)
h
(4.26)
(4.27)
Since µh < µv + QHh ⇒ µh − µv − QHh < 0 and Q0 < Qh then we would be
considering time less than a negative number, which is not possible, so in this
case Q(t) remains below Qh for all (positive) time.
20
Case 5b is when µh > µv +
H
,
Qh
we rewrite Nv (t) ≥ Qh Nh (t) as
−(µv + QH )t
≥ Qh e−µh t
H
Qh
⇒ (µh − µv −
)t ≥ ln
Qh
Q
0
1
Qh
⇒t≤
ln
Q0
(µh − µv − QH )
Q0 e
(4.28)
h
(4.29)
(4.30)
h
When these assumptions are satisfied, the Q(t) will cross above Qh .
In the above cases for Case 5 we start out assuming Q0 < Qh . There are
two possibilities that make this true, either Q0 = Q(0) which is the true initial
condition, or Q0 is the value of Q(t) at some positive point in time when Q(t)
crosses below Qh . In order for the latter case to be true we would have to have
Q0 = Qh by continuity. It works similar for Case 5 with switching the less than
sign to a greater than sign. In summary, if µh < µv + QHh then we end up in Case
5 and if µv + QHh < µh then we end up in Case 4.
Now that we have the conditions, we will plug them into our parameters for
each period and see which case we fall in. Once we get to that point, we use that
case (Case 4 or Case 5) to take limt→∞ Q(t). Once we get to this step, we will
need to compare Q(t) to Qv and Qh for each period to see if we use Case 2 or
Case 3 to calculate the values for the IRNs.
When taking the limit, we either use Case 4 or Case 5; for Case 4 we get:
(µh −µv )t
e
−1
(µh −µv )t
(4.31)
lim Q(t) = Q0 e
+H
t→∞
µv − µh
H
H
,
(4.32)
= Q0 +
e(µh −µv )t +
µv − µh
µh − µv
So with µh > µv , then limt→∞ Q(t) → ∞.
For Case 5 we get:
Nv (t)
lim Q(t) = lim
= lim
t→∞
t→∞ Nh (t)
t→∞
−(µ +
H
)
Nv0 e v Qh t
Nh0 e−µh t
!
(µh −µv − QH )t
= lim Q0 e
h
t→∞
(4.33)
If µh < µv +
H
Q0
⇒ µh − µv −
(µh −µv − QH )t
lim Q0 e
t→∞
h
H
Qh
< 0, than
= 0.
(4.34)
So in summary, if we take the limit for Case 4 we get limt→∞ Q(t) = ∞ which
is definitely greater than Qh and Qv so we use the formula for Case 2. Taking
the limit for Case 5 we get limt→∞ Q(t) = 0 which is less than Qh and Qv so we
use the formula for Case 3.
To sum up all of the sub cases we just covered, if rh , rv = 0 then when
H
µh > µv + max(Q
we have Q(t) → ∞ because the hosts are dying at a faster
0 ,Qh )
rate than the vectors. Otherwise, we have Q(t) → 0.
21
Table 6: Results for which strain wins with the IRN formulas illustrated with a 1 when strain 1
wins and 2 when strain 2 wins with the square-wave model. These are results for the invasion
reproductive number.
Period
Days
R/S
W/S
W/G
1
1-35
2
1
1
2
36-77
2
1
1
4.2.2
3
78-94
2
1
1
4
95-121
2
2
1
5
122-166
2
1
1
6
167-182
1
2
2
7
183-218
1
2
2
8
219-277
2
2
1
9
278-304
2
2
2
10
305-349
2
2
2
11
350-365
2
1
1
Results
We will have two different values for the reproductive numbers. In the Mathematica code, we define the parameters to be seasonally varying. From these
parameters we will get two different values for Q(t). One set of results will be
from solving the system from (3.5) and getting results for the state variables.
From these results we will get values for Q(t) at each moment in time, and we
will plug them into the IRN formulas. This gives the replacement numbers as
a description of what is actually happening at each moment in time [13]. The
second set of values for Q(t) will come from using the parameter definitions and
solving for Nh∗ and Nv∗ using:
µh (t)
∗
,
(4.35)
Nh = Kh 1 −
rh (t)
µv (t)
∗
.
(4.36)
Nv = Kv 1 −
rv (t)
These values for Q(t) will give an equilibrium-based instantaneous IRNs that are
a description of the current ”environment” created by shifting the parameters.
The replacement numbers are more realistic because instead of assuming the
infected outsider is in the population the entire infectious period and mixing with
the population exactly the same way a normal host or vector would, replacement
number is ”the average number of secondary infections produced by a typical
infective during the entire period of infectiousness” [13]. Below is a table that
shows the results of the IRN calculations broken up for each period using the
equilibrium values calculated for each eleven periods.
Table 6 is just the direct calculations using the formula, and equilibrium
values. The below results for the square and sine-wave models were taken from
Mathematica after solving the system and calculating the IRN values along with
analyzing the graphs of the hosts and vectors infected with each strain once
the population settles down. When looking at the IRN graphs in the results,
it is possible to have strain 1 and strain 2 ”win” at different times of the year.
Even though both strains show a clear oscillation, one strain eventually ends
up oscillating around an endemic value, while the other oscillates close to 0. In
the entire process of each strain oscillating from the beginning of the simulation,
the strains never interchange positions, even if we start both strains off at equal
prevalences. Meaning, both strains initial value at t = 0, is equal, but the strain
that will ”win”, always has a higher prevalence value than the ”losing” strain for
the entire simulation, and continuously get further apart. This shows seasonality
is not enough to support coexistence in our model.
22
Host population
Vector population
0.084
125
0.082
0.080
120
0.078
115
0.076
0.074
110
998.2
998.4
998.6
998.8
999.0
(a) Host population graphed for one year period.
998.2
998.4
998.6
998.8
999.0
(b) Vector population graphed for one year period.
120
1.5
100
80
1.0
60
40
0.5
20
0.2
0.4
0.6
0.8
1.0
(c) Growth rate for hosts graphed for one year period.
0.2
0.4
0.6
0.8
1.0
(d) Growth rate for vectors graphed for one year period.
Figure 2: Comparison of total population versus corresponding birth rates with the R/S cycle and
square-wave model.
For raccoon and T. sanguisuga cycle, we compare the total vector population
with the total host population for a one year period. The host population decreases from days 0 to 35, and 219 to 365 while increasing in the middle. Those
days directly coincide with host growth. The vector population decreases from
days 0 to 123, and 219 to 365 while increasing in the middle. Again, those days
directly coincide with vector growth. The results are shown in Figure 2. The
other two cycles behave similarly.
One important comparison to analyze is how seasonality affects competition.
In Figure 3, we illustrate the ties that each strain has with the IRN value with
the W/G cycle over a period of one year. When plotting each strain prevalence
separately, both strains increase and decrease at the same time. The reason for
this is, infection is highly dependent on vector feeding, so when vector feeding is
at its high, each strain will increase. Instead of prevalence, we look at prevalence
gain by subtracting one prevalence from the other. When looking at the gain of
each strain, you can see that strain 1 gains on strain 2 around when the IRN
value for strain 1 is greater than 1. Also, it is shown that strain 2 is gaining
ground for roughly half of the period, yet it still loses out overall. The results
are similar for the other cycles.
Another important result is that seasonality does not support coexistence. As
seen in Figure 4, for all the cycles, one strain always wins while the other goes to
0. Based on the results of Figure 4, we took one model and applied extreme values
to the parameters to test whether seasonality does support coexistence or not.
We wanted to make sure that our parameters were not the reason coexistence
was not occurring. We know that strain 1 heavily favors vector feeding and strain
2 favors host growth and host feeding.
In order to test our model we took a piecewise function, and for the beginning
half of the year we used parameters that favored strain 1, and for the second half
23
IRN
Prevalence gain
1.2
0.40
1.1
0.35
0.2
0.4
0.6
0.8
1.0
0.30
0.9
0.25
0.8
998.2
(a) IRNs R̃1 (solid curve), R̃2 (dashed curve)
998.4
998.6
998.8
999.0
(b) Strain 1 prevalence gain in hosts.
Figure 3: Comparison of the IRN graph with the prevalence gain for strain 2 with the W/G cycle and
square-wave model; the results are similar for hosts. Strain 1 prevalence gain is the exact opposite as
strain 2.
Strain 1 VS Strain 2
0.460
0.455
0.450
0.445
0.440
50
100
150
200
250
Plot of entire range of each strain to illustrate divergence, strain 2 is going to 0.
Figure 4: Comparison of each strain for the vector population for the W/G cycle and the square-wave
model, host population graph is similar.
of the year we used values that favored strain 2. We made sure each half of the
year parameters had approximately the same IRN value, by testing them in a
constant model, so one strain would not dominate. We applied the piecewise
parameters to our model for the W/G cycle since, in reality, both strains have
been found in woodrat populations [22].
The results showed the prevalence values to be very close to equal, but when
we graph the results over the entire time span, it is shown that one strain goes
towards the endemic value, while the other goes to 0. This means that, even
though both strains seem to be sticking around, one strain slowly heads towards
and endemic value, while the other is approaching 0. This implies, in the seasonality based model, after enough time, one strain will always ”win”, while the
other will approach 0. These results are shown in Figure 5. Also shown in Figure
5, each strain increases and decreases at the same time of the year.
The first half of the year, strain 1 is favored, and the second half strain 2
is favored. In reality, strain 1’s strength depends on stercorarian transmission.
This makes strain 1 favored when vector feeding is high, so strain 2 being favored
requires vector feeding to be low. With vector feeding being the only way for
vectors to be infected, this means the vector feeding rate affects overall infection
rates and levels, as well as which strain is advantaged. This explains why each
strain is increasing and decreasing the same times of the year.
24
Strain 1
Strain 2
0.50
0.49
0.49
0.48
0.48
0.47
0.47
0.46
0.46
0.45
0.45
0.44
99.2
99.4
99.6
99.8
100.0
99.2
(a) Prevalence strain 1.
99.4
99.6
99.8
100.0
(b) Prevalence strain 2.
0.4505
0.441
0.4500
0.440
0.4495
0.439
0.4490
0.438
0.4485
0.437
0.4480
0.436
0.4475
20
40
60
80
100
20
(c) Strain 1 over entire range.
40
60
80
100
(d) Strain 2 over entire range.
Figure 5: Prevalence among vectors of W/G test (square-wave) model with extreme parameter
values to see if seasonality does support coexistence. Prevalence for hosts gives similar results.
Since we had this interesting result of both strains increasing and decreasing
together, we investigated it. What we did was, use the same parameters, meaning
strain 1 favored for the first half of the period, and strain 2 favored the second
half, but we made the periods 40 years. When the periods were made longer,
you could see strain 1 headed towards equilibrium, while strain 2 heads towards
0. When the periods are only 1 year, to difference is not as noticeable. With
the transient effects, both strains initially rise together because the first half of
the period favors strain 1, which has the high vector feeding rate and infection
alone really depends on vector feeding. When the periods are short enough, the
turning around may not happen before parameters change again, so the initial
boost dominates during that half-period. The graphs of the 40 year periods are
shown in Figure 6.
In order for strain 2 to be favored, vector feeding had to be less than or equal
to 0.05 (vectors/yr). With a vector feeding level that low, both strains would
decrease. In a model where strain 2 wins, the prevalence value will always be
lower than if strain 1 wins because the vector feeding level is significantly lower
than if strain 1 wins.
25
40 year period
1 year period
0.4
0.50
0.3
0.49
0.48
0.2
0.47
0.1
0.46
0.0
140
160
180
200
10.5
(a) Strain 1 vector prevalence.
11.0
11.5
12.0
(b) Strain 1 vector prevalence.
0.6
0.49
0.5
0.48
0.4
0.3
0.47
0.2
0.46
0.1
0.45
140
160
180
10.5
200
(c) Strain 2 vector prevalence.
11.0
11.5
12.0
(d) Strain 2 vector prevalence.
Figure 6: Graphs displaying difference in transient effects with different periods with the extreme
parameters in the W/G cycle and square-wave model.
4.3
Sine-wave results
For the raccoon and T. sanguisuga cycle we have the following results. In comparison to the total vector population versus the total host population, we will
analyze the increase and decrease phases for one period. For the R/S cycle,
growth and mortality max phases are approximately the same time of year, this
yields results that are not intuitive. You can see populations decreasing the same
time growth increases. The reason for this is mortality starts increasing sooner,
so it takes longer for the population increase to take place from growth rate increase. The graphs of the above qualitative analysis are shown in Figure 7. The
other two cycles display similar results.
26
Host population
Vector population
128
0.081
0.080
126
0.079
124
0.078
122
0.077
0.076
120
998.2
998.4
998.6
998.8
998.2
999.0
(a) Host population graphed for one year period.
998.4
998.6
998.8
999.0
(b) Vector population graphed for one year period.
120
1.5
100
80
1.0
60
40
0.5
20
998.2
998.4
998.6
998.8
998.2
999.0
(c) Growth rate for hosts graphed for one year period.
998.4
998.6
998.8
999.0
(d) Growth rate for vectors graphed for one year period.
0.40
0.55
0.35
0.50
0.45
0.30
0.40
0.25
0.35
0.20
0.30
0.15
0.25
998.2
998.4
998.6
998.8
999.0
(e) Mortality rate for hosts graphed for one year period.
998.2
998.4
998.6
998.8
999.0
(f) Mortality rate for vectors graphed for one year period.
Figure 7: Comparison of total population versus most influential demographic parameter with R/S
cycle and the sine-wave model.
Now we will take the results from the square-wave analysis and compare
them to the sine-wave results. For the W/S cycle we will look at the IRN graphs
with the prevalence gains. The square-wave and continuous graphs are similar.
You can see that the square-wave is more extreme due to the fact that the
parameter values are either at the max or the min, there is no in between, while
the continuous is gradual so you do not have the sharp curves, as seen in Figure
8. The results are similar for the W/G cycle. Also, again both strains increase
and decrease the same time of year. The reasoning is the same as the squarewave model, when strain 1 is being favored, both strains increase due to the high
vector feeding, and when strain 2 is being favored, both strains decrease due to
low vector feeding.
When comparing the continuous versus square-wave for the R/S cycle we get
different results. As seen in Figure 9, the square-wave IRN graph never has strain
1 winning, but the continuous does. This is because the square parameters, when
equal to 0 is not enough for the values to rise above 1 for this particular cycle.
When you look at the replacement number graphs, you can see that strain 1
does ”win” for a small portion of the year because replacement numbers are
actually using the population numbers at that moment in time, instead of using
population equilibrium. Which is why replacement numbers are more realistic.
27
Square-wave
Continuous
1.2
1.2
1.1
1.1
0.2
0.4
0.6
0.8
0.2
1.0
0.9
0.9
0.8
0.8
(a) IRNs R̃1 (solid curve), R̃2 (dashed curve)
0.4
0.6
0.8
1.0
(b) IRNs R̃1 (solid curve), R̃2 (dashed curve)
0.046
0.052
0.044
0.050
0.042
0.048
0.046
0.040
0.044
0.038
0.042
0.036
0.040
0.034
999.2
999.4
999.6
999.8
1000.0
(c) (Ih1 − Ih2 )/Nh , prevalence gain strain 1 over strain 2
999.2
999.4
999.6
999.8
1000.0
(d) (Ih1 − Ih2 )/Nh , prevalence gain strain 1 over strain 2
0.97
1.00
0.96
0.95
0.95
0.90
0.94
0.85
0.93
0.80
0.92
0.75
999.2
999.4
999.6
999.8
1000.0
(e) (Iv1 − Iv2 )/Nv , prevalence of strain 1 over strain 2
999.2
999.4
999.6
999.8
1000.0
(f) (Iv1 − Iv2 )/Nv , prevalence gain strain 1 over strain 2
Figure 8: Comparison of interstrain competition results for square-wave (left column) and continuous (right column) models, over the course of one year for the W/S cycle.
28
Square-wave
Continuous
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
0.2
0.4
0.6
0.8
0.2
1.0
0.9
0.9
0.8
0.8
0.7
0.7
(a) IRNs R̃1 (solid), R̃2 (dashed)
1.4
1.3
1.3
1.2
1.2
1.1
1.1
0.4
0.6
0.8
0.6
0.8
1.0
(b) IRNs R̃1 (solid), R̃2 (dashed)
1.4
0.2
0.4
0.2
1.0
0.9
0.9
0.8
0.8
0.7
0.7
(c) Replacement numbers R̃1 (solid), R̃2 (dashed)
0.4
0.6
0.8
1.0
(d) Replacement numbers R̃1 (solid), R̃2 (dashed)
0.00046
0.000405
0.000400
0.00044
0.000395
0.000390
0.00042
0.000385
0.00040
0.000380
0.000375
999.2
999.4
999.6
999.8
999.2
1000.0
(e) (Ih2 − Ih1 )/Nh , prevalence gain strain 2 over strain 1
999.4
999.6
999.8
1000.0
(f) (Ih2 − Ih1 )/Nh , prevalence gain strain 2 over strain 1
0.69
0.68
0.80
0.67
0.75
0.66
0.70
0.65
0.64
0.65
999.2
999.4
999.6
999.8
1000.0
(g) (Iv2 − Iv1 )/Nv , prevalence of strain 2 over strain 1
999.2
999.4
999.6
999.8
1000.0
(h) (Iv2 − Iv1 )/Nv , prevalence gain strain 2 over strain 1
Figure 9: Comparison of interstrain competition results for square-wave (left column) and continuous (right column) models, over the course of one year with the R/S cycle.
29
5
Conclusion
In the literature review [17] states seasonality can support coexistence. They
consider one insect with two discrete, non-overlapping generations each year. In
their model, they use variables for parameter values and construct the analysis.
They also define two parameters, one is which is basically a measure of seasonality, and s which is the geometric mean of two variables a and b, which are
used to define . Upon completing their analysis they discover that with s =
3.5, and = 0.008, that seasonality does support coexistence. In our model, we
consider two insects, and two hosts with overlapping generations. Based on these
differences, this would explain why our model does not support coexistence.
The vector density for any of the models never reach the observed annual
average density. When we change the parameters so the vector growth is never
at a value of zero, the density jumps up significantly. This implies that when
you truly have periods of no vector growth, it drops the average annual density
around 3%. This is the idea as explained in Figure 1. Even though the piecewise
averages match the averages from [22], when the vector population goes through
the minimum phase, it is not enough to recover back to the population equilibrium found in [22]. When seasonality is applied with the sine-wave model, the
population values are still slightly less than the equilibrium value. This is for the
same reason, when the low is 0, it really effects the population, not making it
possible to completely recover to the equilibrium value.
An interesting finding was that, even though there is a clear ”winner” in each
cycle, both strains are increasing and decreasing at the same time of year. Upon
tests with piecewise defined parameter models with longer periods such as 40
years, it is found that vector feeding plays an important role in the spread of
infection. From using the longer periods, it makes it very obvious that with the
shorter periods, transient behavior dominates when seasonality is put into the
model. Fast enough switching between two parameter sets could make it all about
transient effects by not giving the system enough time to recover before switching
again. From the results, we found the most influential seasonal parameter for
infection is vector feeding. If that gets below a certain threshold, infection dies
out all together.
One of the main questions for this project was to determine if seasonality
makes it possible for two strains to coexist. [22] supports competitive exclusion,
but in reality both strains are found in the woodrat population. One of the points
in the hypothesis was to explain these findings in nature, does seasonality make
it possible for the ”losing” strain to remain in very small numbers due to certain
times of year favoring the ”losing” strain by demographic parameters such as
birth, mortality, and feeding levels to reach max and mins different times of year.
For all of the cycles, in either model, there does not seem to be a summer
”winning” strain and a winter ”winning” strain. As shown in the results, seasonality does not support coexistence. This implies that temporal heterogeneity
alone cannot support coexistence, so spatial heterogeneity and/or local stochasticity must be necessary. This means that time is not enough to create 2 separate
niches: spatial/environmental heterogeneity is required as well. Our model supports a slow competitive exclusion in which the loser recedes over the span of
centuries.
Spatial heterogeneity and/or local stochasticity is important in explaining the
coexistence of two strains. We used the model to apply temporal heterogeneity,
30
but did not include the spatial aspect or local randomness. Based on the results,
a future study idea could be to apply seasonality to the model along with spatial
heterogeneity and local randomness to see if that explains why, in reality, both
strains are being found in the woodrat population. In the end, a lot of interesting
results, but we still have not verified why, in reality, both strains are being
discovered. Based on the results of using seasonality in the model, one strain
should die out completely after enough time. Because of this, further study is
needed, as before, spatial heterogeneity and/or local stochasticity is needed in
the model as well as seasonality.
Acknowledgments
The authors thank Vadim Ponomarenko for suggesting the form of f (t) in Section
3.3. This work was partially supported by the National Science Foundation under
Grant DMS-1020880.
31
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